Cosmology Eta Bexact Rung Derivation

The baryon rung

The baryon-to-photon ratio, conventionally written η_B, is a cosmological quantity: it compares the number of protons and neutrons left over from the early universe to the number of photons in the cosmic microwave background. Its measured value is about six parts per billion. In Recognition Science, this ratio is not a free parameter but a consequence of spatial dimension. The framework's library proves that when space has three dimensions, the integer that sets the ratio's scale is -44, and it derives that integer by three separate routes that must agree.

The first route starts from the framework's account of a recognition event, a discrete record of an interaction. In three spatial dimensions, the event carries D + 2 independent degrees of freedom (three spatial, one temporal, one ledger balance) and D² independent ledger parities. Their product, 3² × 5 = 45, is called the consciousness gap. The rung exponent is one minus that gap: 1 - 45 = -44. The library also proves what happens in other dimensions: in one dimension the rung would be -2, in two dimensions -15, in five dimensions -174.

The second route uses a Gray code cycle on the three-dimensional structure, whose flip counts are [4, 2, 2], and a torsion spectrum from the framework's filtration, whose first gap is 11. Their product, 4 × 11 = 44, negated, gives the same -44. The third route counts the Standard Model's fermionic degrees of freedom, 90, halves them to 45, and subtracts that from one. The library proves all three routes agree, and that the chirality route is only defined at D = 3.

In Recognition Science, the framework models η_B as φ^(-44), where φ is the golden ratio, placing it on a φ-power ladder. The convergence theorem is not a statistical claim; the routes share structural ingredients, and the framework can be falsified at any named input. The library's certificate bundles the three route theorems and their pairwise agreements into a single structure, and the derivation uses no framework-specific axioms, depending only on the ambient type theory's standard postulates.

/-- **THE η_B EXACT RUNG THEOREM**:

    The φ-rung exponent −44 of the baryon-to-photon ratio is forced
    by D = 3 via convergent structural routes that all agree.

    No route uses the empirical value of η_B as input. The agreement
    is a derivation, not a fit. -/
theorem etaBExactRungCert : EtaBExactRungCert where
  route_A_dimension := eta_B_rung_from_dimension_at_D3
  route_B_chirality := eta_B_rung_from_chirality_eq
  route_C_fermionic := eta_B_rung_from_fermionic_eq
  AB_agree := routes_AB_agree
  AC_agree := routes_AC_agree
  BC_agree := routes_BC_agree
  chirality_gap_bridge := chirality_product_equals_gap_minus_one
  fermionic_gap_bridge := fermionic_half_equals_gap
  matches_existing := matches_existing_eta_B_rung
  rung_sum := derived_rung_sum

/-- **Counterfactual at D = 1**: gap-from-dimension would give η_B rung
    = 1 − 1²×3 = 1 − 3 = −2 (very different from observed). -/
theorem D1_counterfactual_rung :
    eta_B_rung_from_dimension 1 = -2 := by
  unfold eta_B_rung_from_dimension Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.A
  decide

/-- **Counterfactual at D = 2**: gap-from-dimension would give η_B rung
    = 1 − 4×4 = 1 − 16 = −15. -/
theorem D2_counterfactual_rung :
    eta_B_rung_from_dimension 2 = -15 := by
  unfold eta_B_rung_from_dimension Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.A
  decide

/-- **Counterfactual at D = 5**: gap-from-dimension would give η_B rung
    = 1 − 25×7 = 1 − 175 = −174. -/
theorem D5_counterfactual_rung :
    eta_B_rung_from_dimension 5 = -174 := by
  unfold eta_B_rung_from_dimension Foundation.GapDerivation.consciousnessGap Foundation.GapDerivation.A
  decide

/-- The chirality route yields −44. -/
theorem eta_B_rung_from_chirality_eq : eta_B_rung_from_chirality = -44 := by
  unfold eta_B_rung_from_chirality
  have hflip : bitFlipCount 0 = 4 := bit0_flips_four
  have htor : (torsionGap 0 1).natAbs = 11 := by native_decide
  rw [hflip, htor]
  decide

/-- D = 3 is the unique non-degenerate dimension where the chirality
    product (4 × 11) and the gap-from-dimension formula (D²(D+2) − 1)
    both produce the same integer 44, because:
    - D = 3 is forced by T8 (linking + 8-tick + sync)
    - The Gray code on Q^D is defined for D = 3 specifically
    - The torsion spectrum {0, 11, 17} is a CW-filtration consequence at D = 3
    For any other D the chirality route does not even type-check
    (bitFlipCount is `Fin 3 → ℕ` by construction). -/
theorem chirality_only_defined_at_D3 :
    eta_B_rung_from_chirality
      = eta_B_rung_from_dimension Foundation.GapDerivation.D := by
  rw [routes_AB_agree.symm]

/-- The derived rung matches the existing definition `BaryonAsymmetryExact.eta_B_rung`. -/
theorem matches_existing_eta_B_rung :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = eta_B_rung := by
  rw [eta_B_rung_from_dimension_at_D3]
  rfl

What this page does not claim

This page does not claim that the framework derives the measured numerical value of η_B, only the integer rung -44. This page does not claim that the three routes are statistically independent; they share structural ingredients. This page does not claim that the framework's derivation is a physical measurement of η_B.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/EtaBExactRungDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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